The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W_{0} (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W_{−1}. The minimum value of x is at {−1/e,−1}
For each integer k there is one branch, denoted by W_{k}(z), which is a complex-valued function of one complex argument. W_{0} is known as the principal branch. These functions have the following property: if z and w are any complex numbers, then
$we^{w}=z$
holds if and only if
$w=W_{k}(z)\ \ {\text{ for some integer }}k.$
When dealing with real numbers only, the two branches W_{0} and W_{−1} suffice: for real numbers x and y the equation
$ye^{y}=x$
can be solved for y only if x ≥ −1/e; we get y = W_{0}(x) if x ≥ 0 and the two values y = W_{0}(x) and y = W_{−1}(x) if −1/e ≤ x < 0.
Main branch of the Lambert W function in the complex plane, plotted with domain coloring. Note the branch cut along the negative real axis, ending at −1/e.
The modulus of the principal branch of the Lambert W function, colored according to argW(z)
The notation convention chosen here (with W_{0} and W_{−1}) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth.^{[2]}
The name "product logarithm" can be understood as this: Since the inverse function of f(w) = e^{w} is called the logarithm, it makes sense to call the inverse "function" of the productwe^{w} as "product logarithm". (Technical note: as it is multivalued, it is not actually a function and thus W is described as the converse relation rather than inverse function.) It is related to the Omega constant, which is equal to W_{0}(1).
History
Lambert first considered the related Lambert's Transcendental Equation in 1758,^{[3]} which led to an article by Leonhard Euler in 1783^{[4]} that discussed the special case of we^{w}.
The function Lambert considered was
$x=x^{m}+q.$
Euler transformed this equation into the form
$x^{a}-x^{b}=(a-b)cx^{a+b}.$
Both authors derived a series solution for their equations.
Once Euler had solved this equation, he considered the case a = b. Taking limits, he derived the equation
$\ln x=cx^{a}.$
He then put a = 1 and obtained a convergent series solution for the resulting equation, expressing x in terms of c.
After taking derivatives with respect to x and some manipulation, the standard form of the Lambert function is obtained.
In 1993, it was reported^{[by whom?]} that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics. In response, Rob Corless and developers of the Maple computer algebra system made a library search and found that this function was ubiquitous in nature.^{[clarification needed]}^{[2]}^{[5]}
Although it was widely believed that the Lambert W function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.^{[6]}
Elementary properties, branches and range
The range of the W function, showing all branches. The black curves (including the real axis) form the image of the real axis, the orange curves are the image of the imaginary axis. The purple curve is the image of a small circle around the point z = 0; the red curves are the image of a small circle around the point z = −1/e.
Plot of the imaginary part of W[n,x+i y] for branches n=-2,-1,0,1,2. The plot is similar to that of the multivalued complex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different
There are countably many branches of the W function, denoted by W_{k}(z), for integer k; W_{0}(z) being the main (or principal) branch. W_{0}(z) is defined for all complex numbers z while W_{k}(z) with k ≠ 0 is defined for all non-zero z. We have W_{0}(0) = 0 and limz→0W_{k}(z) = −∞ for all k ≠ 0.
The branch point for the principal branch is at z = −1/e, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W_{−1} and W_{1}. In all branches W_{k} with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.
The functions W_{k}(z), k ∈ Z are all injective and their ranges are disjoint. The range of the entire multivalued function W is the complex plane. The image of the real axis is the union of the real axis and the quadratrix of Hippias, the parametric curve w = −t cot t + it.
Inverse
Regions of the complex plane for which $W(n,ze^{z})=z$, where z = x + iy. The darker boundaries of a particular region are included in the lighter region of the same color. The point at {−1, 0} is included in both the $n=-1$ (blue) region and the $n=0$ (gray) region. Horizontal grid lines are in multiples of π.
The range plot above also delineates the regions in the complex plane where the simple inverse relationship $W(n,ze^{z})=z$ is true. f = ze^{z} implies that there exists an n such that $z=W(n,f)=W(n,ze^{z})$, where n depends upon the value of z. The value of the integer n changes abruptly when ze^{z} is at the branch cut of $W(n,ze^{z})$, which means that ze^{z} ≤ 0, except for $n=0$ where it is ze^{z} ≤ −1/e.
Defining $z=x+iy$, where x and y are real, and expressing e^{z} in polar coordinates, it is seen that
For $n\neq 0$, the branch cut for $W(n,ze^{z})$ is the non-positive real axis, so that
$x\sin y+y\cos y=0\Rightarrow x=-y/\tan(y),$
and
$(x\cos y-y\sin y)e^{x}\leq 0.$
For $n=0$, the branch cut for $W[n,ze^{z}]$ is the real axis with $-\infty <z\leq -1/e$, so that the inequality becomes
$(x\cos y-y\sin y)e^{x}\leq -1/e.$
Inside the regions bounded by the above, there are no discontinuous changes in $W(n,ze^{z})$, and those regions specify where the W function is simply invertible, i.e. $W(n,ze^{z})=z$.
(The last equation is more common in the literature but is undefined at x = 0). One consequence of this (using the fact that W_{0}(e) = 1) is the identity
where L_{1} = ln x, L_{2} = ln ln x, and [^{l + m} _{l + 1}] is a non-negative Stirling number of the first kind.^{[2]} Keeping only the first two terms of the expansion,
$W_{0}(x)=\ln x-\ln \ln x+o(1).$
The other real branch, W_{−1}, defined in the interval [−1/e, 0), has an approximation of the same form as x approaches zero, with in this case L_{1} = ln(−x) and L_{2} = ln(−ln(−x)).^{[2]}
Integer and complex powers
Integer powers of W_{0} also admit simple Taylor (or Laurent) series expansions at zero:
which is, in general, a Laurent series of order r. Equivalently, the latter can be written in the form of a Taylor expansion of powers of W_{0}(x) / x:
for every $y>1/e$ and $x\geq -1/e$, with equality only for $x=y\log(y)$.
The bound allows many other bounds to be made, such as taking $y=x+1$ which gives the bound
Note that, since f(x) = xe^{x} is not injective, it does not always hold that W(f(x)) = x, much like with the inverse trigonometric functions. For fixed x < 0 and x ≠ −1, the equation xe^{x} = ye^{y} has two solutions in y, one of which is of course y = x. Then, for i = 0 and x < −1, as well as for i = −1 and x ∈ (−1, 0), y = W_{i}(xe^{x}) is the other solution.
For any nonzero algebraic numberx, W(x) is a transcendental number. Indeed, if W(x) is zero, then x must be zero as well, and if W(x) is nonzero and algebraic, then by the Lindemann–Weierstrass theorem, e^{W(x)} must be transcendental, implying that x = W(x)e^{W(x)} must also be transcendental.
The following are special values of the principal branch:
The third identity may be derived from the second by making the substitution u = x^{−2} and the first can also be derived from the third by the substitution z = 1/√2 tan x.
Except for z along the branch cut (−∞, −1/e] (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:^{[17]}
The Lambert W function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the form ze^{z} = w and then to solve for z. using the W function.
For example, the equation
$3^{x}=2x+2$
(where x is an unknown real number) can be solved by rewriting it as
where a, b, and c are complex constants, with b and c not equal to zero, and the W function is of any integer order.
Viscous flows
Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:
where H(x) is the debris flow height, x is the channel downstream position, L is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.
In pipe flow, the Lambert W function is part of the explicit formulation of the Colebrook equation for finding the Darcy friction factor. This factor is used to determine the pressure drop through a straight run of pipe when the flow is turbulent.^{[18]}
Neuroimaging
The Lambert W function was employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain voxel, to the corresponding blood oxygenation level dependent (BOLD) signal.^{[19]}
Chemical engineering
The Lambert W function was employed in the field of chemical engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage. The Lambert W function turned out to be the exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.^{[20]}^{[21]}
Materials science
The Lambert W function was employed in the field of epitaxial film growth for the determination of the critical dislocation onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert W for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert W turns it in an explicit equation for analytical handling with ease.^{[22]}
Porous media
The Lambert W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the lighter fluid.^{[23]}
This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.^{[24]}
Statistics
The centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence (also called the Jeffreys divergence ^{[25]}) has a closed form using the Lambert W function.^{[26]}
Pooling of tests for infectious diseases
Solving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert W function. ^{[27]}^{[28]}^{[29]}
Exact solutions of the Schrödinger equation
The Lambert W function appears in a quantum-mechanical potential, which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to^{[30]}
$z=W\left(e^{-{\frac {x}{\sigma }}}\right).$
The Lambert W function also appears in the exact solution for the bound state energy of the one dimensional Schrödinger equation with a Double Delta Potential.
The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.^{[31]}
Thermodynamic equilibrium
If a reaction involves reactants and products having heat capacities that are constant with temperature then the equilibrium constant K obeys
$\ln K={\frac {a}{T}}+b+c\ln T$
for some constants a, b, and c. When c (equal to ΔC_{p}/R) is not zero we can find the value or values of T where K equals a given value as follows, where we use L for ln T.
If a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly −1/e). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.
Phase separation of polymer mixtures
In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the Edmond-Ogston model, the solutions for binodal and tie-lines are formulated in terms of Lambert W functions.^{[32]}
Wien's displacement law in a D-dimensional universe
Wien's displacement law is expressed as $\nu _{\max }/T=\alpha =\mathrm {const}$. With $x=h\nu _{\max }/k_{\mathrm {B} }T$ and $d\rho _{T}\left(x\right)/dx=0$, where $\rho _{T}$ is the spectral energy energy density, one finds $e^{-x}=1-{\frac {x}{D}}$. The solution $x=D+W\left(-De^{-D}\right)$ shows that the spectral energy density is dependent on the dimensionality of the universe.^{[33]}
AdS/CFT correspondence
The classical finite-size corrections to the dispersion relations of giant magnons, single spikes and GKP strings can be expressed in terms of the Lambert W function.^{[34]}^{[35]}
Epidemiology
In the t → ∞ limit of the SIR model, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.^{[36]}
Determination of the time of flight of a projectile
The total time of the journey of a projectile which experiences air resistance proportional to its velocity can be determined in exact form by using the Lambert W function.
Electromagnetic surface wave propagation
The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation like u ln u = v (where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function. The first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.^{[37]}
Generalizations
The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:
$e^{-cx}=a_{0}(x-r)$
(1)
where a_{0}, c and r are real constants. The solution is
Generalizations of the Lambert W function^{[38]}^{[39]}^{[40]} include:
An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a link (unknown prior to 2007^{[41]}) between these two areas, where the right-hand side of (1) is replaced by a quadratic polynomial in x:
where r_{1} and r_{2} are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument x but the terms like r_{i} and a_{0} are parameters of that function. In this respect, the generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r_{1} = r_{2}, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function. Equation (2) expresses the equation governing the dilaton field, from which is derived the metric of the R = T or lineal two-body gravity problem in 1 + 1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical double-well Dirac delta function model for unequal charges in one dimension.
Analytical solutions of the eigenenergies of a special case of the quantum mechanical three-body problem, namely the (three-dimensional) hydrogen molecule-ion.^{[42]} Here the right-hand side of (1) is replaced by a ratio of infinite order polynomials in x:
where r_{i} and s_{i} are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation (3) with its specialized cases expressed in (1) and (2) is related to a large class of delay differential equations. G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).^{[43]}
Applications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in (1) as seen recently in the area of atomic, molecular, and optical physics.^{[44]}
Plots
Plots of the Lambert W function on the complex plane
z = Re(W_{0}(x + iy))
z = Im(W_{0}(x + iy))
z = |W_{0}(x + iy)|
Superimposition of the previous three plots
Numerical evaluation
The W function may be approximated using Newton's method, with successive approximations to w = W(z) (so z = we^{w}) being
The Lambert W function is implemented as LambertW in Maple,^{[45]}lambertw in GP (and glambertW in PARI), lambertw in Matlab,^{[46]} also lambertw in Octave with the specfun package, as lambert_w in Maxima,^{[47]} as ProductLog (with a silent alias LambertW) in Mathematica,^{[48]} as lambertw in Python scipy's special function package,^{[49]} as LambertW in Perl's ntheory module,^{[50]} and as gsl_sf_lambert_W0, gsl_sf_lambert_Wm1 functions in the special functions section of the GNU Scientific Library (GSL). In the Boost C++ libraries, the calls are lambert_w0, lambert_wm1, lambert_w0_prime, and lambert_wm1_prime. In R, the Lambert W function is implemented as the lambertW0 and lambertWm1 functions in the lamW package.^{[51]}
A C++ code for all the branches of the complex Lambert W function is available on the homepage of István Mező.^{[52]}
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Veberic, D., "Having Fun with Lambert W(x) Function" arXiv:1003.1628 (2010); Veberic, D. (2012). "Lambert W function for applications in physics". Computer Physics Communications. 183 (12): 2622–2628. arXiv:1209.0735. Bibcode:2012CoPhC.183.2622V. doi:10.1016/j.cpc.2012.07.008. S2CID 315088.
Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. arXiv:1601.04895. doi:10.1109/LCOMM.2013.070113.130972. S2CID 10062685.
External links
Wikimedia Commons has media related to Lambert W function.
National Institute of Science and Technology Digital Library – Lambert W
MathWorld – Lambert W-Function
Computing the Lambert W function
Corless et al. Notes about Lambert W research
GPL C++ implementation with Halley's and Fritsch's iteration.
Special Functions of the GNU Scientific Library – GSL